1. Pricing of Discrete Path-Dependent Options by the Double-Exponential Fast Gauss Transform method Yusaku Yamamoto Nagoya University CORS/INFORMS International Meeting May 18, 2004

2. Outline of the talk 1. Introduction 2. The DE formula and the fast Gauss transform 3. Pricing of discrete barrier options 4. Pricing of discrete lookback options 5. Pricing of CDD temperature derivatives 6. Conclusion

3. 1. Introduction Pricing of path-dependent options –Consider a path dependent option whose payoff function h depends on the whole history of the asset price S t until maturity: –The rational price of this option can be computed as a discounted expectation value (under the risk-neutral measure) of the payoff function: Discrete path-dependent options –In this talk, we consider the special cases where h depends on the asset prices at discrete times t 0 = 0, t 1, …, t n = T: Q 0 (S 0 ) = e –rT E 0 [h({S t })]. h = h (S 0, S 1, …, S n ). h = h ({S t | 0 < t < T }).

4. Examples of discrete path-dependent options Discrete barrier options (down-and-out call) –Same as European call options except that the option is nullified if the asset price falls below the pre-specified barrier level H. Discrete lookback put options –The option holder can sell the asset at the highest price the asset took between the initial and maturity dates. CDD temperature derivatives (call) max (S n – K, 0) if S i > H for all 0 < i < n, 0 otherwise. h = h = max (S 0, S 1, …, S n ) – S n h = max (CDD – K, 0), where CDD =  max (T i – T, 0) n StSt t T K H StSt t T nullified payoff TiTi n T

5. Computational approach Pricing by a recursion formula –None of the above three options have explicit formulas for Q 0 (S 0 ). –Monte Carlo method is widely used. But convergence is very slow. –Instead, their prices can be calculated using recursion formulas. Example: discrete barrier options –Let P i (S i ) be the probability density that the option is still alive at time t i and the asset price at t i is S i. –Then P i (S i ) satisfies the following recursion formula: –where p(S i |S i–1 ) is the transition probability density function of S i. –The option price can be computed by

6. Computational approach (cont’d) Reduction to convolution –For the Black Scholes and related models, the t.p.d.f. of the log asset price x t can be represented by Gaussian distribution and the recursion formula reduces to convolution of a function with the Gaussian as follows: Lookback options and CDD derivatives –Similar techniques can be applied to discrete lookback options and CDD temperature derivatives (under the so-called Dischel temperature model) as well and the pricing problems can be reduced to a series of convolutions.

7. Efficient computation of the convolutions Existing methods –Tridiagonal probability algorithm (Tse et al., 2001) –Convolution method (Reiner, 2000) Our approach (Broadie and Yamamoto, 2004) –Combination of the double-exponential integration formula and the fast Gauss transform Comparison of computational work and accuracy Method Tse et al. Reiner Ours Numerical methods employed Gaussian quadrature Simpson’s formula + FFT DE formula + FGT Work O(N2)O(N2) O(NlogN) O(N)O(N) Error O(exp(–cN)) O(N–d)O(N–d) O(exp(–cN)/logN) N: the number of sample points at each time step

8. 2. The DE formula and the fast Gauss transform The double-exponential integration formula –The convolution requires integration of an analytic function over a half-infinite line: –The double exponential formula (Takahashi & Mori, 1974) is efficient for this type of integral. Main idea of the DE formula –Transform the integral into an integral of a rapidly decaying function over the entire real axis using the change of variables: –Apply the trapezoidal rule to evaluate the latter. –It is well known that the error of the trapezoidal rule decreases exponentially with N (number of sample points) in this case.

9. The double exponential formula (cont’d) The integral after the change of variables Weights and sample points of the DE formula –Note that the sample points are not equally spaced in the x space.

10. The fast Gauss transform Motivation –By applying the DE formula, the convolution becomes discrete convolution as follows: –Direct evaluation of this convolution requires O(N 2 ) work. –We cannot use the FFT to reduce the work to O(NlogN) because the sample points a j i is not equally spaced. –However, by exploiting the fact that p G (x) is Gaussian, we can use the fast Gauss transform (Greengard & Strain, 1991) and reduce the computational work for each convolution to O(N).

11. Main idea of the fast Gauss transform Convolution to be computed –Suppose that we want to compute the discrete convolution: Expansion of Gaussian by Hermite functions –To compute the sums efficiently, we use the following expansion of the Gaussian probability density function: –This can be shown easily using the expansion –and the definition of the Hermite function. ( h n (x): Hermite function ) i i

12. Algorithm A three step algorithm –Truncating this expansion, we have –This suggests a three step approach to calculate the sums of Gaussians (Greengard & Strain, 1991): –(1) Compute. (O(N) work) –(2) Multiply the result by and sum over n. (O(1) work) –(3) Multiply the result byand sum over m. (O(N) work) –Thus we can compute G(x i ) (i=1, …, N) in O(N) work. i i

13. Main idea of the fast Gauss transform t t +1 yjyj y0y0 xixi x0x0

14. 3. Pricing of discrete barrier options Target problem –Discrete down-and-out call options under the Black-Scholes model Numerical methods –Reiner’s method (Simpson’s formula + FFT) –Our method (DE formula + FGT) Computational environment –Pentium II PC with Red-Hat Linux –gnu++ compiler Parameters –S 0 = K = 100, r = 0.1, q = 0,  = 0.3, T = 0.2, H = 91, 99 (2) –Number of monitoring dates = 5, 25 and 50

15. Numerical results for barrier options absolute error Time (sec) The convergence of our method is much faster than that of Reiner’s method. It can compute the price of down-and-out call options with 50 monitoring dates within 0.3 seconds up to accuracy of 10 -10.

16. 4. Pricing of discrete lookback options Reduction to a 1-dimensional problem –The price of a lookback put option can be expressed as follows: –By applying a change of measure (Andreasen, 1998) this can be rewritten as – By introducing log stock prices by s i = log(S i /S 0 ) and m i =log(M i /S 0 ), –So we only need the probability density function of m n – s n. where,.

17. Pricing of discrete lookback options (cont’d) The recurrence formula –The pdf of m n – s n has the form of c i  (x) + g i (x) and it can be shown that c i and g i (x) satisfy the following recurrence (Broadie and Yamamoto, 2004). –This is again a convolution of a function with Gaussian and can be computed efficiently using the DE formula and the fast Gauss transform. where.

18. Numerical experiments Target problem –Discrete lookback put options under the Black-Scholes model Numerical methods –Tse et al.’s method (Gaussian quadrature) –Our method (DE formula + FGT) Parameters –S 0 = 100, r = 0.1, q = 0,  = 0.3, T = 0.5 –Number of monitoring dates = 5, 25 and 50

19. Numerical results for discrete lookback options absolute error Time (sec) Our method is orders of magnitude faster than Tse et al.’s method. It can compute the price of knockout options with 50 monitoring dates within 0.3 seconds up to accuracy of 10 -8. Monitoring dates n=5n=25n=50 Tridiagonal method * 0.66s20.55s442.48s Our approach 0.008s0.098s0.30s Computational time for absolute error < 10 -8 * From Tse et al. (2001)

20. 5. Pricing of CDD temperature derivatives Problem formulation –The price of a CDD call derivative is expressed as follows: –If we define and denote the joint probability density function of T k and C k by p k (T k, C k ), then we can rewrite the price as h = max (CDD – K, 0), CDD =  max (T i – T, 0) n Q 0 (T 0 ) = e –rT E 0 [h] where C k = ∑ i=1 k max(0, T i – T ) max(C n – K, 0) p n (T n, C n ).

21. Pricing of CDD derivatives (cont’d) The joint transition probability density of T k and C k –In the so-called Dischel model, the transition probability density of the temperature can be written as –When T k > T, C k = C k–1 + (T k – T ) and the joint transition probability density can be computed as –The joint tpdf can be computed similarly when T k < T. kk k k k kk k k k kk kk k k kkk （ ） k k k

22. Pricing of CDD derivatives (cont’d) The recursion formula –Finally, when T k > T, the recursion formula for p k (T k, C k ) can be written as which is a convolution of a function with Gaussian and can be computed by our method efficiently. –When T k < T, p k (T k, C k ) can be computed by a similar recursion formula. kk k k kkkkkkk k k k k k k k k,

23. Numerical experiments Target problem –CDD call derivatives under the Dischel model Numerical methods –Monte Carlo method –Our method Parameters –Period of observation: N days from July 7 th (N = 10 or 20) –Place of observation: Tokyo –Index: CDD (T = 24 C) –Strike value: K= 20 or 40 C –  = – 0.56,  = – 0.01,  = 1.83,  k = 20 o o

24. Numerical results Time (sec) Price N=10, K=20N=20, K=40 The MC method needs 100 to 1000 seconds to get an accuracy of 10 –2. Our method is more than 10 times faster than the MC method.

25. 6. Conclusion Under the Black-Scholes and related frameworks, the price of many path-dependent options such as the discrete barrier options, discrete lookback options and CDD temperature derivatives can be computed by a series of convolutions of a function with the Gaussian distribution. We proposed a new algorithm to compute these convolutions efficiently by a combination of the DE integration formula and the fast Gauss transform. The numerical experiments show that our method is much faster and much more accurate than existing methods in computing the prices of the above path-dependent options.

26. For a complete version of the paper, please contact me at yamamoto@na.cse.nagoya-u.ac.jp